FRFs (indicated by dark gray shading). As can be seen, the rotated elliptic envelope gives less conservative estimation of the uncertainties.
Figure 4.5(c) shows the stability lobes corresponding to the averaged FRF and to the measured FRFs (ten times) for low spindle speeds. The lower envelope of the stability lobes corresponding to the measured individual FRFs are indicated by gray shading. This region can be called the estimated robust stable domain: within this region, the system is stable for all the ten FRFs. It can be seen that the averaged FRF strongly overestimates the robustly stable region.
Figure 4.5(d) shows the same averaged stability boundaries and the estimated robust stability boundaries together with the robust stability boundaries obtained by the circular and the rotated elliptic concepts. It can be seen that both the circular and the elliptic envelopes provide almost the same curve as the estimated robust stability boundary. Since the fitted tubes comprises the measured FRFs, the estimation of the robust boundary is always conservative, i.e., the circular and the elliptic estimations are below the experimental robust stability boundary.
In Fig. 4.5(e), the robust stability boundaries are shown for high spindle speeds. In this case, the structure of the robust stability boundaries are more segmented than in the low spindle speed region. The robust stability estimations still provide a good approximation of the lower envelope.
The analysis demonstrates that the estimated robust stability boundary can well be bounded by the proposed method. The results also confirms that the robust stability region can significantly be smaller than the stability region of the averaged model. This observation may explain the mismatch between stability lobe predictions and actual cutting tests, which is often experienced in machine tool chatter research.
4.5 Main results
Stability lobe diagrams of turning operations in the plane on machining parameters can be cal-culated based on the frequency response function of the tool. I have modeled the uncertainty of measurements and modal parameters by an envelope centered around the nominal FRF. Based on the cross-section of the envelopes, I have derived the following conditions to calculate the robust stability boundaries.
Thesis statement 3
A conservative estimation on the robustness of turning operations in case of static uncertainty of the frequency response functions (FRF) can be given based on an envelope centered around the nominal frequency response function. Let the nominal complex FRF be denoted byH(ω) = HRe(ω) + iHIm(ω), moreover, let the envelope be determined by a rotated elliptical cross section with main and minor axes w1(ω), w2(ω), and letα(ω)be the angle between the main axis of the ellipse and the real axis. The robust machining parameters can be determined with the formula of the safety factor
SFc= min
ω≥0
H˜cr,Re(ω) cos α(ω)
−H˜cr,Im(ω) sin α(ω) w1(ω)
!2
+
H˜cr,Re(ω) sin α(ω)
+ ˜Hcr,Im(ω) cos α(ω) w2(ω)
!2!1/2
,
46 CHAPTER 4 Turning operations where
H˜cr,Re(ω) = − 1
2κ −HRe(ω), H˜cr,Im(ω) = sin(ωτ)
2κ 1−cos(ωτ) −HIm(ω)
denote the critical complex perturbation,τ = 60/Ωis the regenerative time delay, Ωis the speed of rotation of the workpiece given in rpm, andκis the specific cutting force coefficient. Any pair of machining parameters(Ω∗, κ∗)are guaranteed to be robust against bounded static uncertainty in the frequency response functions, if the system without uncertainties is stable andSFc>1.
Related publications: [114], [115].
CHAPTER 5
Robust control design for turning operations
In order to extend the domain of stable parameters and suppress machine tool chatter effectively, several engineering applications have been proposed. For instance, a vibration absorber is presented in [116], an impedance modulation technique in [117] and a self-tuning dynamic vibration absorber in [118]. An adaptive chatter suppression technique is proposed in [119], which is based on a real-time chatter detection and automatic spindle speed adjustment. A Pyragas-type feedback control is presented in [120] to mitigate chatter in high-speed milling. A chatter control is also introduced in [121] based on the structured singular values, where the spindle speed and depth of cut are treated as uncertain parameters in order to enlarge the domain of stable machining parameters. A PD-controller-based chatter suppression method for turning operations is presented in [122], where actuation and measurement are utilized theoretically at the tool tip.
In this chapter we investigate a robust PD controller for turning operations with feedback delay, where uncertainties arise in the frequency response functions. The inaccurate measurement of the frequency response function matrix between the actuation point and tool tip can significantly affect the stability of the controlled system. In order to find the domain of robust stable control gains, the so-called structured singular values (µ-values) have been calculated.
Section 5.1 introduces the model of turning operations extended with a PD controller. Con-sidering the uncertainty of the FRF matrix, the robust stability analysis is carried out in Sec. 5.2.
Based on measured data, a case study is presented in Sec. 5.3. Results are concluded in Sec. 5.4.
5.1 Model of turning operations with control
The dynamical model of turning operations including feedback control is presented in Fig. 5.1.
The cutting forceF1(t)acting at the tool tip (associated with coordinateq1(t)) is determined by the formula
F1(t) =wfq h(t)
, (5.1)
wherefq(h)is the specific cutting force defined by characteristics introduced in (4.2),wis the depth of cut and h(t)is the instantaneous chip thickness, see [103]. Due to the vibrations of the tool, the chip thickness is determined not only by the feed motion, but also by the current and previous positions of the tool one revolution before. For constant spindle speeds, the regenerative time delay can be given explicitly asτ1 = 60/Ω, whereΩis the workpiece revolution given in rpm. Then the modified cutting force is given as
F1(t) = wfq vfτ1+q1(t−τ1)−q1(t)
, (5.2)
wherevf is the feed velocity, andvfτ1 =h0is the prescribed chip thickness.
As opposed to Chapter 4, here we apply also a controller acting at coordinateq2(t)to stabilize the arising vibrations. When the position and velocity are measured at the actuation point, the
47
48 CHAPTER 5 Robust control design for turning operations
(a)
(b) (c)
PC
Tool Workpiece
h(t)
vf
w
Ω
0 0
0 0
0 0
F1(t)
F2(t) F2(t)
Fd,1(t)
Fd,2(t) q1(t)
q2(t)
x1(t) x1(t)
x2(t) x2(t)
qst,1
qst,2
x y z
Fig. 5.1. Model of turning operations with feedback control: (a) Dynamical model; (b) Non-deformed shape; (c) Deformed shape (equilibrium state).
actuator force can be given by the feedback law
F2(t) =−kpq2(t−τ2)−kdq˙2(t−τ2), (5.3) wherekpandkdare the proportional and the derivative control gains andτ2is the inherent feedback delay in the control loop, which originates from the finite time of information propagation, sampling period, analog-to-digital conversion etc.
In order to simplify derivations, the following notation is used F(t) =
F1(t) F2(t)
and q(t) = q1(t)
q2(t)
. (5.4)
The frequency response function matrixH(ω)between the forcingF(ω) = F(F(t))and displace-mentQ(ω) =F(q(t))is defined as
H(ω)F(ω) =Q(ω), (5.5) which in this case can be expressed as
H11(ω) H12(ω) H21(ω) H22(ω)
F1(ω) F2(ω)
=
Q1(ω) Q2(ω)
. (5.6)
H11(ω)andH22(ω)are the direct frequency response functions andH12(ω)andH21(ω)are the cross frequency response functions, respectively. Although, the cross termsH12(ω)andH21(ω)are equal theoretically, they are never exactly the same in real measurements. A measured frequency response functions matrix is presented in Fig. 5.2, which shows the data from 25 single measurements (gray curves) and their average (black curves). It can be seen, that the uncertainty of the cross terms is slightly larger than the direct terms. It can also be observed that the individual measurements of the cross terms provide similar output, however, H12(ω) andH21(ω)are different. This observation
5.1 Model of turning operations with control 49
ω[Hz]
ω[Hz]
ω[Hz]
ω[Hz]
0 0
0 0
1000 1000
1000 1000
2000 2000
2000 2000
3000 3000
3000 3000
4000 4000
4000 4000
5000 5000
5000 5000
10−5 10−5
10−5 10−5
10−6 10−6
10−6 10−6
10−7 10−7
10−7 10−7
10−8 10−8
10−8 10−8
10−9 10−9
10−9 10−9
|H11(ω)|[m/N] |H12(ω)|[m/N]
|H21(ω)|[m/N] |H22(ω)|[m/N] SingleSingle
Single Single
Average Average
Average Average
Fig. 5.2.Measured frequency response function matrix of the tool.
indicates that the mounting of the sensors and direction of excitations strongly affect the measured FRFs. These uncertainties can be included in the robust analysis.
During the stability analysis, the static parts of the forcing can be separated, and only the perturbed motion has to be considered. Assuming that the position vector can be written as q(t) = qst+x(t), whereqstis the static deformation andx(t)is perturbation about the equilibrium, the forcing can be decomposed to static and dynamic terms as
F(t) =Fst+Fd(t), (5.7) where
Fst=
wfq h0
−kpqst,2
(5.8)
and
Fd(t) =
wfq0(h0) x1(t−τ1)−x1(t)
−kpx2(t−τ2)−kdx˙2(t−τ2)
. (5.9)
The static force Fst results in a static deformation qst that can be determined from the relation qst =H(0)Fst. Then the stability of the variational system around the static equilibriumqst can be analyzed.
The Fourier transform of (5.9) with some algebraic manipulations gives Fd(ω) =
wfq0(h0)X1(ω) e−iωτ1 −1
−(kp+kdiω)X2(ω)e−iωτ2
=
wfq0(h0) e−iωτ1 −1
0
0 −(kp+kdiω) e−iωτ2
| {z }
=: K(ω)
X(ω), (5.10)
50 CHAPTER 5 Robust control design for turning operations whereFd(ω) =F(Fd(t))andX(ω) =F(x(t)). Substitution of (5.10) into (5.5) and simplification with the static terms yields
H(ω)K(ω)X(ω) =X(ω), (5.11) which can be rearranged as
I−H(ω)K(ω)
X(ω) =0. (5.12) The existence of a periodic solution about the static equilibriumqstimplies that
G(ωc;w,Ω) := det I−H(ωc)K(ωc)
= 0, (5.13)
where ωc is the chatter frequency, the frequency of the arising vibrations. In order to construct stability lobe diagrams in the plane(Ω, w), (5.13) can be considered as a co-dimension one problem, where the real and imaginary parts ofG(ωc;w,Ω)give two scalar equations and a one-dimensional curve is sought in the parameter space ofΩ, w andωc. The multi-dimensional bisection method developed by [104] is an efficient and fast numerical tool for this task.
Note, that the transition curves calculated from the condition (5.13) separate the space of the machining parameters, where the number of unstable characteristic exponents is constant, but does not identify the stable regions. Identification of stable domains require the application of the Cauchy’s argument principle [6, 123]. The number P of poles located on the open right half complex plane can be determined by the formula
P = 1 2πi
Z ∞
−∞
G0(ωc;·)
G(ωc;·)dωc = 1 πi
Z ∞
0
G0(ωc;·)
G(ωc;·)dωc, (5.14) where G0(ωc;·) denotes differentiation of G(ωc;·) with respect toωc (for details, see [6]). This number can be approximated by numerical integration by taking the upper limit up to sufficiently large chatter frequencies.